In every state of this country and throughout the world, underground aquifers and other subterranean groundwaters have been contaminated. The procedures utilized to clean up these water resources are expensive; hence, it has become necessary to optimize the methods by which these natural resources can be made purer.
The current invention entails a method of pumping the contaminants out of the underground waters. The method has developed an optimized procedure that performs well, even when substantial uncertainty exists regarding aquifer characteristics.
The invention uses a "weighted feedback" law that adjusts pumping rates of extraction wells, when the observed values of hydraulic head and pollution concentration deviate from the predicted values over time.
Several approaches have been proposed in the prior art to preserve the value of optimization analysis, when there is significant error in the aquifer model, i.e., changes over time. One such method.sup.21 is a chance-constrained technique for finding constant pumping policies to remediate contaminated aquifers. The method used the inverse problem via nonlinear multiple regression to obtain parameter estimates and to estimate the covariance matrix for the uncertain flow and transport parameters. These values were used in a chance-constrained algorithm, using first-order first- and second-moment analysis to obtain an optimal strategy with an estimate of the probability that the aquifer will be cleaned. This method does not allow for the use of new information obtained during the remediation to alter the policy. The policy is completely determined before the remediation even begins.
Another method.sup.20 presents a neural network technique for identifying the most pessimistic or constrained realizations of hydraulic conductivity values in a set generated randomly for use in a chance-constrained optimization. The procedure employs hydraulic gradient control strategies for contaminant capture. This method is tested to identify pessimistic realizations for a linear programming optimization model of gradient control for a two-dimensional aquifer. This approach can allow large sets of conductivity realizations to be used in chance-constrained methods, because nonconstraining conductivity realizations can be screened before the optimization in order to reduce computational complexity.
This neural network screening method.sup.19 has further been applied to a mixed integer chance-constrained programming model for hydraulic gradient control. For this newer technique, it is crucial to reduce the number of constraints corresponding to conductivity realizations, because the computational effort needed to solve the problem grows geometrically with the number of constraints.
Another proposed method.sup.4 reduces parameter uncertainty as a function of future measurements, remediating the contaminated aquifer in a cost-effective manner. The optimization problem is resolved each time new parameter estimates are obtained. This method is compared to an unweighted feedback law generated by differential dynamic programming, neglecting second derivatives of the transition function and using the active constraint method. The method has been tested on a 20-node, one-dimensional aquifer. It does not incorporate inequality constraints. This method has been applied to a two-dimensional aquifer; the approach has been extended to include more general constraints. A feedback law using the penalty function method or second-derivative information from the transition equation has not been used, however, with this technique.
A further method.sup.22 uses what is called a stochastic programming technique, with recourse to calculate optimal steady-state pumping policies for gradient contaminant containment. The cost of plume escape or recourse cost is included in this objective. Four model formulations were solved, using a linear finite difference approximation for the steady-state flow equations. The stochastic programming approach finds the least-cost policy constrained to contain the modeled pollution plume, using both the best-guess parameter values and a specified set of different realizations of the uncertain parameters. Violation of the containment constraints is possible, if recourse costs are included. Recourse costs are the costs associated with the escape of the plume from the containment region; they include fines and the costs associated with drilling new wells. Policies calculated by these methods will satisfy the containment constraint over a wide range of conditions (typically, by pumping a great deal).
The inventive method herein represents a different approach to incorporating uncertainty than has been utilized in most earlier approaches. The feedback procedure developed by the current inventors uses input probability distributions for uncertain parameters in a different way than do way other, former approaches. Frequently, information is inadequate to reliably determine probability distributions. Feedback laws can be tested using a range of probability distributions. In particular, feedback laws can be developed and tested without the assumption that the analyst can assign relative probabilities to each set of randomly generated parameters. This property allows the analyst to develop feedback policies which are robust, with respect to assumed probability distributions used to characterize model uncertainty and error.
Another way to view this property is to recognize that feedback policies developed by the weighted feedback approach are more sensitive to measurements taken during the remediation than they are to assumed distributions. This relieves the analyst of the task of producing an "accurate" probability distribution.
Unlike some earlier approaches, the inventive feedback policies use information during the remediation as it becomes available; the policies are not assumed to be completely determined before the remediation begins (as in earlier methods.sup.21).
Unlike the foregoing procedure.sup.4 that reduces parameter uncertainty as a function of future measurements, the invention takes into account feedback laws generated by the penalty function method, coupled with differential dynamic programming as the basis of a nonlinear feedback law. Feedback laws generated in this way are sensitive to all constraints.